CN111832200A - A Frequency Response Analysis Method for Cyclic Symmetrical Structures with Additional Dry Friction Dampers - Google Patents

Abstract

The invention discloses and belongs to the technical field of structural vibration reduction mechanics calculation method, in particular to a cyclic symmetric structure frequency response analysis method with additional dry friction dampers, and the specific operation of the cyclic symmetric structure frequency response analysis method with additional dry friction dampers The steps are as follows: S1: establish the dynamic equation of the structural member and damper sector, S2: establish the finite element model of the sector, obtain the mass and stiffness matrix of the sector, S3: apply the modal synthesis method to compress the degrees of freedom of the sector model, S4: Add complex constraint boundary conditions to the compressed model, S5: Time domain representation of dry friction force, S6: Apply hybrid time-frequency method to solve nonlinear equations, this kind of additional dry friction damper frequency response analysis method for cyclic symmetrical structure , greatly saves storage resources and computing time, and the computing time saving ratio is equivalent to the saving ratio of storage resources, reducing the number of nonlinear degrees of freedom involved in iteration, effectively reducing the risk of convergence, and forming an efficient and robust frequency response algorithm.

Description

A Frequency Response Analysis Method for Cyclic Symmetrical Structures with Additional Dry Friction Dampers

technical field

The invention relates to the technical field of structural vibration reduction mechanics calculation methods, in particular to a cyclic symmetrical structure frequency response analysis method with additional dry friction dampers.

Background technique

In the aviation industry, rotating structural parts are usually made into thin-walled structures. Thin-walled structures are prone to overall large-scale vibration under the excitation of dynamic loads, resulting in large dynamic stress, resulting in premature initiation of fatigue cracks and sudden breakage in a specific frequency range. The failure of power or transmission key components will cause accidents. The introduction of dry friction can significantly suppress the vibration amplitude of thin-walled rotating components. effect to achieve the purpose of vibration reduction. In order to maintain the dynamic balance of the rotating structure, the dry friction damper should also be designed as a simple axisymmetric structure or a cyclic symmetrical structure, so that the dry friction damper will form a new cyclic symmetrical combined structure together with the rotating structure.

Although the structure of the dry friction damper is simple, it is difficult to design a dry friction damper with good vibration damping performance, mainly including: (1) The dry friction damper is a displacement-dependent damper, that is, its damping effect is mainly It depends on the vibration displacement of the structural parts, and this damping effect affects the vibration history of the structure. The two are coupled with each other. Once the relative displacement of the friction surface cannot be calculated, the dry friction force, damping effect, frequency response, energy consumption, viscosity- The slip ratio and structural dynamic stress cannot be obtained, which means that the design of the damper has no basis; (2) the dependence of the dry friction force on the vibration displacement is a strong nonlinear dependence, which means that the steady state solution needs to be obtained by The iterative method is completed, and the convergence of the iterative method is always the key to whether the calculation can be completed; (3) The modern design method emphasizes the high fidelity of the analysis model in order to accurately describe the details of the analyzed object, which inevitably involves more It also means that more friction contacts need to be set to describe the stick-slip characteristics of different contact parts. The iterative calculation of the frequency response solution process under the participation of multiple degrees of freedom will give dynamics Convergence of the equations brings additional burdens. If the number of degrees of freedom participating in the iteration cannot be effectively reduced, the dynamic equations are likely to be difficult to converge.

At present, the frequency response analysis of additional dry friction dampers is mainly divided into two methods: (1) Equivalent damping method, commercial finite element software (ANSYS, ABAQUS, NASTRAN, MARC, etc.) cannot analyze nonlinear frequency response problems, scholars and Engineers approximate the hysteretic characteristics of dry friction with first-order harmonics, describe its hysteretic characteristics with equivalent damping and equivalent stiffness, and carry out secondary development of commercial finite element software to calculate the structural frequency response, (2) ignoring structural cycles Symmetric characteristics, the sector is directly taken out, and the cyclic symmetry boundary conditions are not applied for calculation. This calculation method ignores the fact that the damping effect of dry friction is not only related to the calculated sector, but also to the adjacent sectors; When the stiffness of the interval has a large coupling, the solution result of this method will have a greater deviation from the real response, or even an error.

SUMMARY OF THE INVENTION

The purpose of the present invention is to provide a frequency response analysis method of a cyclic symmetrical structure with an additional dry friction damper, so as to solve the problems of low precision and inability to save storage resources in the existing method proposed in the above-mentioned background art.

In order to achieve the above purpose, the present invention provides the following technical solutions: a cyclic symmetrical structure frequency response analysis method of an additional dry friction damper, and the specific operation steps of the cyclic symmetrical structure frequency response analysis method of the additional dry friction damper are as follows:

S1: Establish the dynamic equations of the structural member and damper sector:

The dynamic equation of the structure-damper combination sector with the additional dry friction damper is:

where M _G,D is the mass matrix of the structural member and damper sector; K _G,D is the stiffness matrix of the structural member and the damper sector; Q _G,D is the degree of freedom displacement of the structural member and the damper sector; its The previous point and the two points represent the first and second derivatives; F _{CF, G, D} respectively represent the force acting on this sector by the adjacent sector; F _G represents the external excitation force; F _NL represents the dry friction force.

S2: Establish sector finite element model and obtain sector mass and stiffness matrix:

Commercial finite element software (such as ABAQUS, ANSYS, etc.) can be used to establish a sector finite element model, and the mass and stiffness values of the structural sector and damper sector can be derived, and organized into a mass matrix (diagonal) according to the order of nodes and degrees of freedom. matrix) and stiffness matrix, the derived mass and stiffness matrices are M _G , M _D and K _G , K _D in the dynamic model of formula (1). Calculate its mass and stiffness matrix;

If the stiffness matrix is derived from finite element software, it should be noted that the matrix is derived in the Cartesian coordinate system, and there must be a node (left or right) on one side of the sector that does not belong to the sector (belongs to its adjacent sector). area), so it is necessary to transform the stiffness value corresponding to the degree of freedom that does not belong to the sector. The formula is:

In the above formula, T _o is the transformation matrix, and k _ii (i=x, y, z) is the component of stiffness along the x, y, and z directions in the Cartesian coordinate system.

S3: Apply the modal synthesis method to compress the degrees of freedom of the sector model:

Classify the degrees of freedom of the sector model: for the structural sector, keep the friction points, excitation points, vibration pickup points, and the degrees of freedom corresponding to the nodes on the left and right sides of the sector as the main degrees of freedom; for dry friction dampers, keep The node degrees of freedom on the left and right sides of the friction point and the sector are the master degrees of freedom, and the other node degrees of freedom are slave degrees of freedom. The master degrees of freedom are fixed, and the Craig-Bampton method is used to compress the model in the first step:

In the above formula, I is the identity matrix, Q _m is the main degree of freedom, G is the static compression matrix, Φ is the basis vector retained after modal analysis, and is the coordinate corresponding to these retained basis vectors.

S4: Add complex constrained boundary conditions to the compressed model:

Select the nodal degrees of freedom on the left and right sides of the structural member and the damper sector as the master degrees of freedom, and all the other nodal degrees of freedom as slave degrees of freedom, and perform static (Guyan) reduction on the sector model:

In the above formula, Q _m,aux and Q _s,aux are the master and slave degrees of freedom, respectively; G _aux is the statically compressed matrix, similar to the G matrix in formula (3); Q _L and Q _R represent the left, Nodal degrees of freedom on the right side; R _aux is an auxiliary mode, and the rigid matrix and mass matrix corresponding to this auxiliary mode are:

After formulas (4) to (5), the main degrees of freedom are only the degrees of freedom of the left and right boundary nodes, and the degrees of freedom of the auxiliary stiffness and mass matrix are also reduced to the number of main degrees of freedom;

Apply cyclic symmetric boundary conditions to the compressed auxiliary modal mass and auxiliary modal stiffness matrix, perform a modal analysis, and obtain a series of mode shapes, denoted by Ψ, such that the nodes on the left and right sides The motion of the degrees of freedom can be expressed as:

where η _I is the modal coordinate corresponding to the base Ψ. After the above two steps of compression, the nodal degrees of freedom of the original finite element sector model can be written as:

where Γ is the final reduction matrix, which is the product of the transformation matrix in equation (3) and the reduction result of equation (6). The final stiffness matrix and mass matrix can be written as:

S5: Time domain representation of dry friction:

The damping effect of the damper on the structure is achieved through frictional contact. The contact motion state of the damper and the structure can be divided into three situations: sticking, separation and slippage. The following formula gives the contact friction force in the three states The calculation method of F _nl :

S6: Apply the mixed time-frequency method to solve the nonlinear system of equations:

When a structure is subjected to periodic loads, its steady-state response can be considered to be periodic. Therefore, the displacement sequence and external excitation force of the structure can be represented by the Fourier harmonic sequence. It is assumed that n _h harmonics can accurately describe the dynamic response of the structure. , the displacement and excitation force can be written as:

In the formula, ω is the angular frequency of the excitation force, Re represents the real operation, X and F represent the Fourier coefficients of the displacement and the excitation force, k represents the harmonic order, j is the complex root, j*j=-1, similar to , the frictional force f _nl caused by the motion can be written as a function of the displacement sequence:

The above formula shows that the dry friction force is a function of displacement. Substituting formula (10) into formula (11), applying the harmonic balance method (HBM), the complex nonlinear coupling equations of the n _h group can be obtained:

In the above formula, Λ _k =-(kω) ² M+jkωC+K, namely the dynamic stiffness matrix, M, K are the transformed values in formula (8), and the C value is obtained according to the equivalent damping method in step 2, Write equation (12) in the following format:

Θ(u)=Λu- _FFnl (u) (13).

Compared with the prior art, the beneficial effect of the present invention is: under the action of solving the characteristic load of the round, the frequency response calculation method of the cyclic symmetrical structure of the additional dry friction damper, the method considers the cyclic symmetry characteristics of the rotating structural parts, and the dynamic The reduction technology, the modal synthesis method, the round characteristics of the external excitation, and the timely-frequency domain conversion method are comprehensively used to solve the nonlinear equations with strong boundaries. Compared with the frequency response calculation method of the overall model, the algorithm can guarantee the calculation accuracy. On the premise, not only the storage resources are greatly saved (the storage resources are reduced to 2/N, N is the number of sectors forming the cyclic symmetric structure) and the calculation time (equivalent to the saving ratio of the storage resources), but also the number of participating iterations is effectively reduced. The number of nonlinear degrees of freedom effectively reduces the risk of convergence and forms an efficient and robust frequency response solution algorithm.

Description of drawings

Fig. 1 is the schematic diagram of the overall scheme diagram of the algorithm of the present invention;

FIG. 2 is a schematic diagram of the compression of degrees of freedom according to the present invention;

3 is a schematic diagram of a sector finite element model of a cyclically symmetrical structure of the present invention;

FIG. 4 is a schematic flow chart of the calculation flow of the hybrid time-frequency method of the present invention;

FIG. 5 is a schematic diagram of the calculation result of the frequency response of the thin-walled bevel gear with the additional dry friction damper of the present invention.

Detailed ways

The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, but not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

The invention provides a technical solution: a method for analyzing the frequency response of a cyclic symmetric structure with an additional dry friction damper, and the specific operation steps of the method for analyzing the frequency response of a cyclic symmetric structure with an additional dry friction damper are as follows;

A cyclic symmetric structure frequency response analysis method with an additional dry friction damper, and the specific operation steps of the cyclic symmetric structure frequency response analysis method with an additional dry friction damper are as follows:

S1: Establish the dynamic equations of the structural member and damper sector:

The dynamic equation of the structure-damper combination sector with the additional dry friction damper is:

where M _G , M _D , K _G , K _D and C _G , C _D represent the sector mass, stiffness and damping matrix of the structure and damper, respectively; F _G is the external excitation; F _nl represents the action on the object The nonlinear force can be expressed as a function of displacement; Q is the degree of freedom of the structure, which is a column vector, and the upper point and the two points represent the first and second derivatives with respect to time; F _{CF, G, D} respectively represent the adjacent fan The force acting on this sector.

Equation (1) is the dynamic equation of the combined structure sector, and the direct solution of this equation cannot represent the vibration frequency response of the overall structure, and boundary conditions need to be added to it in the subsequent step (step 4);

S2: Establish sector finite element model and obtain sector mass and stiffness matrix:

Commercial finite element software (such as ABAQUS, ANSYS, etc.) can be used to establish a sector finite element model, and the mass and stiffness values of the structural sector and damper sector can be derived, and organized into a mass matrix (diagonal) according to the order of nodes and degrees of freedom. matrix) and stiffness matrix, the derived mass and stiffness matrices are M _G , M _D and K _G , K _D in the dynamic model of formula (1). Calculate its mass and stiffness matrix;

If the stiffness matrix is derived from finite element software, it should be noted that the matrix is derived in the Cartesian coordinate system, and there must be a node (left or right) on one side of the sector that does not belong to the sector (belongs to its adjacent sector). area), so it is necessary to transform the stiffness value corresponding to the degree of freedom that does not belong to the sector. The formula is:

In the formula, K _j is the node located on one side of the sector (left or right); T _o is the transformation matrix; k _ii (i=x, y, z) is the Cartesian coordinate system, the stiffness along x, y, z direction component. μ is the central angle corresponding to each sector, and its value is 2π/N or -2π/N according to the adjacent sector nodes on the left and right sides. The above formula takes the z-axis as the symmetry axis, and the axial direction of the structure is different, so the transformation matrix should also be adjusted accordingly. When transforming, all the corresponding stiffness values in the stiffness matrix involving the nodal degrees of freedom on the side face need to be transformed;

If the stiffness finite element model is prepared by itself, attention should also be paid to the coordinate system in the integration process of the sector stiffness matrix. If it is a cylindrical coordinate system or a spherical coordinate system, no transformation is required. Transform it, for the mass matrix, since it is a diagonal matrix, there is no need to transform it;

For the damping terms C _G and C _D in equation (1), the following forms are equivalent: C _i =αK+βM, i=G, D, take α=2.6*10 ^-5 s, β=7.49s ^-1 ;

So far, the stiffness matrix, mass matrix and damping matrix of the model have been completely prepared, and the model has a large number of degrees of freedom, which is unfavorable for the subsequent calculation amount and iterative convergence, and measures should be taken to compress the degrees of freedom;

S3: Apply the modal synthesis method to compress the degrees of freedom of the sector model:

Classify the degrees of freedom of the sector model: for the structural sector, keep the friction points, excitation points, vibration pickup points, and the degrees of freedom corresponding to the nodes on the left and right sides of the sector as the main degrees of freedom; for dry friction dampers, keep The node degrees of freedom on the left and right sides of the friction point and the sector are the master degrees of freedom, and the other node degrees of freedom are slave degrees of freedom. The master degrees of freedom are fixed, and the Craig-Bampton method is used to compress the model in the first step:

即静态缩减矩阵(Guyan法缩减矩阵)；Φ为约束模态，η_s为对应于Φ的模态坐标，通常约束模态及其对应坐标仅取前几阶即可，这样可在满足求解精度的条件下，忽略大部分不可能出现的高阶模态振型，实现了自由度压缩，即用少数几阶模态振型和其对应的模态坐标表达了所有从自由度可能出现的振动状态；R_CB即Craig-Bampton缩减矩阵。注意：R_CB模态中含有刚体模态，应将其删除，产生刚体模态的原因是未对扇区模型其他节点加约束；In the formula, Q _m is the master degree of freedom; Q _s is the slave degree of freedom; I is the identity matrix, the dimension of which is equal to the number of master degrees of freedom;

That is, the static reduction matrix (Guyan method reduction matrix); Φ is the constraint mode, η _s is the modal coordinate corresponding to Φ, usually the constraint mode and its corresponding coordinates only take the first few orders, which can meet the accuracy of the solution. Under the condition of , ignoring most of the impossible high-order modal shapes, the compression of degrees of freedom is realized, that is, all possible vibration states from the degrees of freedom are expressed by a few modal shapes and their corresponding modal coordinates; R _CB stands for Craig-Bampton reduction matrix. Note: R _CB mode contains rigid body mode, which should be deleted. The reason for generating rigid body mode is that other nodes of the sector model are not constrained;

So far, by using the Craig-Bampton method, the first compression of the degrees of freedom of the sector model is completed;

S4: Add complex constrained boundary conditions to the compressed model:

The number of degrees of freedom of the structural member-damper sector model is greatly reduced after the compression in the previous step, but there are still two problems: (1) The number of degrees of freedom is still large, because when the main degree of freedom is selected, the sector Including the degrees of freedom of the nodes on the left and right sides, the number of nodes on these two sides is relatively large; (2) the sector model does not add complex constraint boundary conditions and cannot be used to calculate the frequency response of the entire structure. To this end, this step will address both issues;

Select the nodal degrees of freedom on the left and right sides of the structural member and the damper sector as the master degrees of freedom, and all the other nodal degrees of freedom as slave degrees of freedom, and perform static (Guyan) reduction on the sector model:

In the above formula, Q _m,aux and Q _s,aux are the master and slave degrees of freedom, respectively; G _aux is the statically compressed matrix, similar to the G matrix in formula (3); Q _L and Q _R represent the left, Nodal degrees of freedom on the right side; R _aux is an auxiliary mode, and the rigid matrix and mass matrix corresponding to this auxiliary mode are:

After formulas (4) to (5), the main degrees of freedom are only the degrees of freedom of the left and right boundary nodes, and the degrees of freedom of the auxiliary stiffness and mass matrix are also reduced to the number of main degrees of freedom;

Apply cyclic symmetric boundary conditions to the compressed auxiliary modal mass and auxiliary modal stiffness matrix, perform a modal analysis, and obtain a series of mode shapes, denoted by Ψ, such that the nodes on the left and right sides The motion of the degrees of freedom can be expressed as:

即复约束边界条件，j为单位复数根，上式意义为：在循环对称结构中，其所有扇区都具有相同的振动形式(模态)，不同的是，扇区间存在一个相位差，这个相位差即为

In formula (6)

That is, the complex constraint boundary condition, j is the unit complex root, the meaning of the above formula is: in the cyclic symmetric structure, all its sectors have the same vibration form (modal), the difference is that there is a phase difference between the sectors, this The phase difference is

where η _I is the modal coordinate corresponding to the base Ψ. After the above two steps of compression, the nodal degrees of freedom of the original finite element sector model can be written as:

where Γ is the final reduction matrix, which is the product of the transformation matrix in equation (3) and the reduction result of equation (6). The final stiffness matrix and mass matrix can be written as:

由于自由度压缩过程中将激励施加位置节点对应自由度设置成了主自由度，所以在变换之后，其大小维持原值；For external force terms

Since the degree of freedom corresponding to the node where the excitation is applied is set to the main degree of freedom during the compression of the degree of freedom, its size remains the original value after the transformation;

After the above steps, the reserved degrees of freedom are compressed as little as possible; by adding complex constraints, the sector model is transformed into a form that can represent the overall model, that is, the frequency response of the overall model is obtained by solving the transformed model;

S5: Time domain representation of dry friction:

The damping effect of the damper on the structure is achieved through frictional contact. The contact motion state of the damper and the structure can be divided into three situations: sticking, separation and slippage. The following formula gives the contact friction force in the three states Calculation method:

where K _t is the tangential stiffness of the contact surface (contact point pair); x _G,τ (t) is the tangential DOF displacement of the structural sector on the contact surface; x _D,τ (t) is the damping on the contact surface is the tangential degree of freedom displacement of the actuator sector; μ is the sliding dry friction coefficient, which is a certain value; N(t) is the normal force applied to the contact point pair. The nonlinear force term in the balance equation can be calculated from equation (9);

So far, the mass, damping, stiffness, external excitation (size and displacement), and nonlinear force in the solution model are all available, and the next step is to solve the frequency response of the dynamic equation in a steady state;

S6: Apply the mixed time-frequency method to solve the nonlinear system of equations:

When a structure is subjected to periodic loads, its steady-state response can be considered to be periodic. Therefore, the displacement sequence and external excitation force of the structure can be represented by the Fourier harmonic sequence. It is assumed that n _h harmonics can accurately describe the dynamic response of the structure. , the displacement and excitation force can be written as:

In the formula, ω is the angular frequency of the excitation force, Re represents the real operation, X and F represent the Fourier coefficients of the displacement and the excitation force, k represents the harmonic order, j is the complex root, j*j=-1, similar to , the frictional force f _nl caused by the motion can be written as a function of the displacement sequence:

The above formula shows that the dry friction force is a function of displacement. Substituting formula (10) into formula (11), applying the harmonic balance method (HBM), the complex nonlinear coupling equations of the n _h group can be obtained:

In the above formula, Λ _k =-(kω) ² M+jkωC+K, namely the dynamic stiffness matrix, M, K are the transformed values in formula (8), and the C value is obtained according to the equivalent damping method in step 2, Write equation (12) in the following format:

Θ(u)=Λu- _FFnl (u) (13);

Θ(u) is the nonlinear system of equations to be solved, and u is the displacement harmonic vector (ie, the part X _k ). At the beginning of the solution, the initial value u(0) is given, and a series of calculated values u(1), u(2), ... u(m) are generated by iteration. If the i-th group of values u(i) makes the equation system Θ( u) The result is close enough to 0, that is, the convergence condition is considered to be satisfied. For each set of given u, the iterative process needs to calculate the value of Θ and solve the Jacobian matrix J at the same time. Due to the strong nonlinear characteristics of dry friction, it is difficult to describe it with expressions, so it is difficult to write the theory of the Jacobian matrix. format; the present invention approximates the Jacobian matrix by numerical difference quotient method or Broyden method;

The advantage of the HFT method to calculate the dynamic response of the structure is that the calculation of the nonlinear dry friction force is carried out in the time domain, so the nonlinear dry friction force related to the displacement history can be accurately calculated. At the same time, the time domain calculation can easily obtain the complex dry friction contact. The nonlinear hysteresis loop is beneficial to determine the stick-slip motion of the damper; the iteration in the frequency domain is beneficial to obtain the stable frequency response of the structure;

In the HFT method, the calculation of the Θ(u) value is completed by the transformation of displacement and dry friction in the frequency domain and time domain, which mainly includes the following steps:

(a) Perform Inverse Fast Fourier Transform (IFFT) on u to obtain the displacement history of each friction contact in the sector;

(b) Calculate the dry friction force according to the displacement history obtained in (a);

(c) Perform Fast Fourier Transform (FFT) on the dry friction force sequence to obtain the harmonic coefficients representing the dry friction force, and store them in f _nl (u), where f _nl (u) is a non-linear harmonic matrix of linear forces;

(d) Continue to calculate the value of Γ(u) according to the f _nl (u) obtained in (c), if the criterion does not converge, iteratively generate a new u, and return to (a) to continue the calculation; if it converges, calculate the next frequency point .

If several iterations are performed at some frequency points and the convergence is still not achieved, it may be caused by the convergence criterion, the selection of the initial value of the iteration, the construction of a Jacobian-like matrix, or the excessive number of nonlinear degrees of freedom. Start debugging, and finally achieve convergence.

So far, the frequency response analysis of the cyclic symmetrical structure with dry friction is completed.

The calculation program can guide the design of parameters such as damper configuration, installation prestress and installation position, and finally obtain a dry friction damper with good vibration damping performance.

Based on the above, the frequency response analysis method of the cyclic symmetrical structure of the additional dry friction damper, under the action of solving the characteristic load of the round, the calculation method of the frequency response of the cyclic symmetrical structure of the additional dry friction damper, the method considers the rotating structure. Due to the characteristics of cyclic symmetry, the dynamic reduction technology, the modal synthesis method, the round characteristics of the external excitation, and the time-frequency domain conversion method are comprehensively used to solve the strong boundary nonlinear equation system. Compared with the frequency response calculation method of the overall model, the algorithm not only greatly saves storage resources (storage resources are reduced to 2/N, N is the number of sectors forming a cyclic symmetrical structure) and calculation time under the premise of ensuring the calculation accuracy. (equivalent to the saving ratio of storage resources), and effectively reduce the number of nonlinear degrees of freedom participating in iteration, effectively reduce the risk of convergence, and form an efficient and robust frequency response solution algorithm.

As a supplement to the knowledge of the algorithm process, the following is a brief description of ① the frequency response solution principle of general structures ② the reduction principle of the dynamic equation of cyclic symmetric structures ③ the round characteristics of external excitation.

(1) Frequency response equation of general structure

The "general structure" mentioned here refers to a non-cyclic symmetric structure, or a model that is a cyclic symmetric structure but has not yet considered its cyclic symmetry characteristics. Its dynamic equation is:

Mx&&(t)+Cx&(t)+Kx(t)=f _l (t)+f _nl [x](t) (1)

where M, K, and C represent the mass, stiffness and damping matrices, respectively; f _l is the external periodic excitation; f _nl represents the nonlinear force acting on the object, which can be expressed as a function of displacement; x is the structural degree of freedom, is a column vector.

When a structure is subjected to periodic loads, its steady-state response can also be considered periodic. Therefore, the displacement sequence and external excitation of the structure can be represented by the Fourier sequence. Assuming that n _h harmonics can describe the dynamic response of the structure accurately enough, the displacement and excitation force can be written as:

In the formula, ω is the angular frequency of the excitation force, Re represents the real operation, X _k , F _l,k represent the Fourier coefficients of the displacement and the excitation force, k represents the harmonic order, j is the complex root, j*j= -1, similarly, the nonlinear force f _nl due to motion can be written as a function of the displacement sequence:

Substituting equations (2) to (4) into equation (1) and applying the harmonic balance method (HBM), the complex nonlinear coupling equations of the n _h group can be obtained:

In the above formula, Λ _k =-(kω) ² M+jkωC+K, namely the dynamic stiffness matrix, and k represents the harmonic order.

(2) The principle of cyclic symmetric structure reduction

If the frequency of the excitation term is ω, it is now assumed that n _h Fourier harmonic sequences can accurately represent the dynamic response of the structure, then the displacement, excitation force and dry friction force can be written in complex form:

为克罗内克乘积符号，

则分别表示位移、外激励和摩擦力的第k阶谐波，0表示基本扇区，j为复数根。

is the symbol for the Kronecker product,

Then respectively represent the k-th harmonic of displacement, external excitation and friction, 0 represents the fundamental sector, and j is a complex root.

Introducing the dynamic stiffness matrix formula (5), and applying the harmonic balance method to formula (1), we can get

In the above formula, the sector numbered 0 represents the basic sector, and _ns represents the number of sectors forming a cyclic symmetric structure.

The Fourier matrix is introduced to transform the above formula as follows:

Equation (8) can be transformed into a diagonal format:

应用克罗内克乘积的性质和

的性质，式(11)可约化为：The k-th order dynamic stiffness matrix of the q sector tooth in the above formula is

Apply the properties of the Kronecker product and

The properties of , formula (11) can be reduced to:

In the above formula, q(k,e _o )-ke _o ≡0[n _s ] means: for a cyclic symmetric model composed of n _s sectors, only when q (q is the Fourier matrix parameter) When the product of harmonic order k and excitation order e _o is equal, its value is not 0, and the rest are 0.

So far, the principle of cyclic symmetric structure reduction has been explained. Through the reduction, the sectors that make up the structure are separated from the overall structure. The model, from formula (8) to formula (10), uses the Fourier matrix to transform the overall frequency response equation to the process of diagonalization, and introduces a constraint, called the complex constraint, for details, see "Technical Solutions" "Step 4: Add reduction conditions to the compression model.

(3) The requirements of the round characteristics of the excitation:

Consider the second and third terms on the left-hand side of equation (8), namely F _l,k and F _nl,k . For F _nl,k , the dry friction force, due to the correlation of its displacement history, that is, the friction force is a function of the displacement history, it can be known that when the displacement x is periodic, F _nl,k is also periodic.

For the external load, in addition to satisfying the periodicity in the time domain, it also needs to meet the characteristics of the space periodicity, that is, the external load must satisfy the following two conditions at the same time:

f _l (t)=f _l (t+T) (11)

and

f _l (s)=f _l (s+H) (12)

The above two equations are the cycle characteristics of the load, which means that the load has periodic characteristics in both time and space. In equation (11), t is time, and T is the time period of the load; in equation (12), s is The load acts on the position of the object, H is the position period of the load, and the frequency response analysis method of the present invention can be used only when the load meets the above two equations. This is because if the load does not meet the above conditions, the load term of the dynamic equation will be It cannot be decoupled, that is, the second term of equation (9) cannot be diagonalized.

The above requirements seem harsh, but fortunately, most of the rotating structures in steady-state operation meet the requirements of equations (11) and (12), such as the meshing operation of gears, the blades of compressors, fans, pumps, etc. The excitation force of gas (liquid) meets the periodic characteristics in time-space at the same time: the dynamic load acts on the sectors constituting the structure successively, and the load circulates periodically in the time domain, and even the motion of the steel wheel on the rail can be approximated It also satisfies the requirements of space-time periodicity, so the universality of the algorithm is guaranteed.

Please refer to the attached drawings, "Acquisition of stiffness and mass parameters" in Figure 1: mainly to obtain the stiffness and mass matrix of structural components and damper sectors necessary for the solution. For complex structures, in order to make the calculation high fidelity, it needs to be derived by the finite element method Rigid matrix and mass matrix; for simple structures, the finite element program can be directly written, the grids can be planned, and the stiffness and mass matrix can be calculated. The constraints that need to be imposed when the structure is transformed into a sector model; the "modal synthesis method compresses the linear degrees of freedom", which means that the modal coordinates and the corresponding basis vectors are used to express the linear degrees of freedom, while the nonlinear degrees of freedom are expressed by modal coordinates and corresponding basis vectors. Still under the physical coordinates, "Excitation Order Analysis": mainly examines whether the external excitation constitutes periodic characteristics in time and space, and this module needs to transform the excitation into the frequency domain as the excitation term for the subsequent frequency response equation; in After the friction contact surface is discretized by dividing the finite element mesh, the contact state of the discretized small area is replaced by the contact point, "contact surface tangent, normal stiffness solution and friction model selection": the function is to solve the Normal stiffness and assign stiffness values to the selected model;

Please refer to the attached drawings again. In Fig. 2, most of the degrees of freedom will be transformed to be represented by a few modal bases and corresponding modal coordinates through modal operations; nonlinear degrees of freedom must participate in the time-frequency domain interactive iteration Therefore, its physical coordinates are retained; the "reserved linear degrees of freedom" are the excitation points and a small number of vibration pickup points set for the convenience of observing the calculation results;

Please refer to the attached drawings again. In Figure 3, the finite element model of the cyclically symmetrical structural sector (taking the spur gear with damping ring installed as an example):

(a) The finite element model of the damping ring sector and the nodal degrees of freedom preserved during Craig-Bampton compression;

(b) The sector-tooth finite element model and its degrees of freedom preserved in Craig-Bampton compression;

Please refer to the accompanying drawings again. In Figure 4, the hybrid time-frequency method (HFT method) is used to solve the problem, and the displacement history is used to solve the dry friction force in the time domain; in the frequency domain, the harmonic components of various forces are iteratively solved;

Please refer to the attached drawings again. In Fig. 5, the calculation result of the frequency response of the thin-walled bevel gear with the additional dry friction damper is shown as the amplitude of the frequency response at a certain point on the web of the bevel gear with the damping bowl installed. This calculation example The parameter analysis of the damping bowl gasket thickness (this dimension affects the normal force) provides a useful reference for the damping bowl installation parameters.

Although the present invention has been described above with reference to the embodiments, various modifications may be made and equivalents may be substituted for parts thereof without departing from the scope of the invention. In particular, as long as there is no structural conflict, the various features in the disclosed embodiments of the present invention can be combined with each other in any way, and the description of these combinations is not exhaustive in this specification. For the sake of omitting space and saving resources. Therefore, the present invention is not limited to the specific embodiments disclosed herein, but includes all technical solutions falling within the scope of the claims.

Claims (1)

1. A method for analyzing the frequency response of a circularly symmetric structure of an additional dry friction damper is characterized by comprising the following steps: the method for analyzing the frequency response of the circularly symmetric structure of the additional dry friction damper comprises the following specific operation steps:

s1: building a structural member and damper sector kinetic equation:

the structure of the additional dry friction damper-the dynamic equation of the sector of the damper assembly is as follows:

in the formula, M_G,DA structural member and damper sector mass matrix; k_G,DA structural member and damper sector stiffness matrix; q_G,DThe displacement is the freedom degree displacement of the sector of the structural part and the damper; one and two points above it represent the first and second derivatives; f_CF，G,DRespectively representing the force of the adjacent sectors on the sector; f_GRepresenting an external excitation force; f_NLIndicating dry friction.

S2: establishing a sector finite element model, and acquiring a sector mass and rigidity matrix:

the sector finite element model can be established by commercial finite element software (such as ABAQUS, ANSYS and the like), the mass and rigidity values of the structural sector and the damper sector are derived, the mass and rigidity values are arranged into a mass matrix (diagonal matrix) and a rigidity matrix according to the node sequence and the freedom sequence, and the derived mass and rigidity matrix is M in the dynamic model of the formula (1)_G,M_DAnd K_G,K_DFor a simpler structure, a finite element calculation program code can be self-programmed to calculate the mass and rigidity matrix of the structure;

if the stiffness matrix is derived from finite element software, it should be noted that the matrix is derived from a rectangular coordinate system, and a node (left or right) on one side of a sector does not belong to the sector (belongs to an adjacent sector), so that the stiffness value corresponding to the degree of freedom not belonging to the sector needs to be transformed, and the formula is as follows:

in the above formula, T_oTo convert the matrix, k_ii(i ═ x, y, z) is right angleThe stiffness in the coordinate system is divided into x, y, and z components.

S3: and (3) applying a modal synthesis method to compress the freedom degree of the sector model:

and classifying the freedom degrees of the sector models: for a structural member sector, the degrees of freedom corresponding to a friction point, an excitation point, a vibration pickup point and nodes on the left side surface and the right side surface of the sector are kept as main degrees of freedom; for a dry friction damper, the degrees of freedom of nodes on a friction point and the left side surface and the right side surface of a sector are reserved as main degrees of freedom, the degrees of freedom of other nodes are slave degrees of freedom, the main degrees of freedom are fixed, and a Craig-Bampton method is applied to perform first-step compression on a model:

in the above formula, I is a unit array, Q_mThe principal degree of freedom, G is a static compression matrix, phi is the basis vectors retained after modal analysis, and phi is the coordinates corresponding to these retained basis vectors.

S4: adding complex constraint boundary conditions for the compressed model:

selecting the node degrees of freedom of the left side surface and the right side surface of the sector of the structural member and the damper as a main degree of freedom, and all the other node degrees of freedom as slave degrees of freedom, and performing static (Guyan) reduction on a sector model:

in the above formula, Q_m,auxAnd Q_s,auxRespectively as master and slave degrees of freedom; g_auxI.e., a statically compressed matrix, similar to the G matrix in equation (3); q_LAnd Q_RRespectively representing degrees of freedom of nodes on the left side surface and the right side surface; r_auxFor the auxiliary mode, the rigid matrix and the mass matrix corresponding to the auxiliary mode are as follows:

after the equations (4) - (5), the main degree of freedom only has the degrees of freedom of the left and right boundary nodes, and the scale of the auxiliary rigidity and mass matrix is reduced to the number of the main degree of freedom;

applying a circularly symmetric boundary condition to the compressed auxiliary modal mass and auxiliary modal stiffness matrix, performing modal analysis to obtain a series of vibration modes, and recording the vibration modes as psi, so that the motion of the node degrees of freedom on the left and right side surfaces can be expressed as:

in the formula eta_IFor the modal coordinates corresponding to the base Ψ, the nodal degrees of freedom of the original finite element sector model can be written as:

where the final reduction matrix is the product of the transformation matrix in equation (3) and the reduction result in equation (6), the final stiffness matrix and the quality matrix may be written as:

s5: time domain representation of dry friction:

the vibration reduction effect of the damper on the structure is realized through frictional contact, and the contact motion state of the damper and the structure can be divided into three conditions: the calculation method of the viscosity, the separation and the slippage comprises the following steps:

s6: solving a nonlinear equation set by applying a mixed time-frequency method:

when the structure is under the action of periodic load, the steady-state response of the structure can be considered to be periodic, so that the displacement sequence and the external excitation force of the structure can be represented by Fourier harmonic sequence, and n is assumed to be_hThe individual harmonics may describe the dynamic response of the structure accurately enough,the displacement and excitation forces can be written as:

where ω is the angular frequency of the excitation force, Re is the real operation, X, F are the displacement and fourier coefficients of the excitation force, k is the harmonic order, j is a complex root, j X j is-1, and similarly, the friction force F caused by motion_nlCan be written as a function of the displacement sequence:

the above formula shows that the dry friction is a function of displacement, and n can be obtained by substituting formula (10) for formula (11) and applying Harmonic Balance Method (HBM)_hComplex nonlinear coupled equations of the set:

Λ_kX_k-F_l,k-F_nl,k(X₀,...,X_nh)＝0,(k＝1,L,n_h) (12)；

in the above formula_k＝-(kω)²M + jk ω C + K, i.e. the dynamic stiffness matrix, M, K, i.e. the transformed value in equation (8), C is obtained according to the equivalent damping method in step 2, and equation (12) is written as follows:

Θ(u)＝Λu-F-F_nl(u) (13)。

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